The number of passengers who have boarded a ship is modeled by the differentiable function P, where t is the number of hours since boarding began. Values of P(t) for selected values of t are given in the table above. (a) According to the model, what is the average rate at which passengers board the ship, in passengers per hour, over the time interval 1≤t≤8 hours? (b) Write P′(4.5) as the limit of a difference quotient. Use the data in the table to approximate P′(4.5). Show the computations that lead to your answer. (c) Must there be a time t, for 3≤t≤6, at which P(t)=500 ? Justify your answer. (d) The total number of gallons of water used by the passengers on the ship is modeled by the function G defined by G(t)=120t sqrt(t) for 0≤t≤8, where t is the number of hours since boarding began. Find G′(4), the rate at which passengers use water, in gallons per hour, at time t=4 hours.
Show all of your work, even though the question may not explicitly remind you to do so. Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied. Your work will be scored on the correctness and completeness of your methods as well as your answers. Answers without supporting work will usually not receive credit.
Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If your answer is given as a decimal approximation, it should be correct to three places after the decimal point.
Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
The number of passengers who have boarded a ship is modeled by the
(a) According to the model, what is the average rate at which passengers board the ship, in passengers per hour, over the time interval 1≤t≤8 hours?
(b) Write P′(4.5) as the limit of a difference quotient. Use the data in the table to approximate P′(4.5). Show the computations that lead to your answer.
(c) Must there be a time t, for 3≤t≤6, at which P(t)=500 ? Justify your answer.
(d) The total number of gallons of water used by the passengers on the ship is modeled by the function G defined by G(t)=120t sqrt(t) for 0≤t≤8, where t is the number of hours since boarding began. Find G′(4), the rate at which passengers use water, in gallons per hour, at time t=4 hours.
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